reserve x,x1,x2,x3 for Real;

theorem Th17:
  cos(3*x)=4*(cos(x))|^3-3*cos(x)
proof
  cos(3*x)=cos((x+x)+x) .=cos(2*x)*cos(x)-sin(x+x)*sin(x) by SIN_COS:75
    .=(2*(cos(x))^2-1)*cos(x)-sin(2*x)*sin(x) by Th7
    .=2*(cos(x)*cos(x)*cos(x))-1*cos(x)-sin(2*x)*sin(x)
    .=2*((cos(x))|^1*cos(x)*cos(x))-1*cos(x)-sin(2*x)*sin(x)
    .=2*((cos(x))|^(1+1)*cos(x))-1*cos(x)-sin(2*x)*sin(x) by NEWTON:6
    .=2*((cos(x))|^(2+1))-cos(x)-sin(2*x)*sin(x) by NEWTON:6
    .=2*(cos(x))|^3-cos(x)-(2*sin(x)*cos(x))*sin(x) by Th5
    .=2*(cos(x))|^3-cos(x)-2*cos(x)*(sin(x)*sin(x))
    .=2*(cos(x))|^3-cos(x)-2*cos(x)*(1-cos(x)*cos(x)) by SIN_COS4:4
    .=2*(cos(x))|^3-cos(x)-(2*cos(x)-2*(cos(x)*cos(x)*cos(x)))
    .=2*(cos(x))|^3-cos(x)-(2*cos(x)-2*((cos(x))|^1*cos(x)*cos(x)))
    .=2*(cos(x))|^3-cos(x)-(2*cos(x)-2*((cos(x))|^(1+1)*cos(x))) by NEWTON:6
    .=2*(cos(x))|^3-cos(x)-(2*cos(x)-2*((cos(x))|^(2+1))) by NEWTON:6
    .=2*(cos(x))|^3+2*(cos(x))|^3-3*cos(x);
  hence thesis;
end;
